End-effects caused by a finite aperture

The ideal acquisition pattern for plane-wave synthesis is a fixed-split spread with wide aperture (Figure ). With this acquisition, every receiver will record reflected wave fields caused by every source wave field. In practice, however, a moving one-side spread (Figure ) is the typical acquisition pattern in marine surveys.

The forward model in equation () implicitly assumes an infinite aperture experiment. In order to model correctly, we need to incorporate the aperture effect as follows:

$${\bf g}_j(z_0) = A_j F(z_0,z_0) {\bf s}_j(z_0),$$ (19)

where A_j is a diagonal matrix with 1, where the receiver is located and elsewhere. Since the aperture operator, A_j, changes from source to source in a moving one-side spread, the wave-stack equation () is no longer valid. In order to overcome this problem, we need to fill the missing traces before wave stacking.

In the case of the one-sided spread acquisition, we can expand the acquisition area using the principle of reciprocity, which states that if we exchange the source and receiver positions, we would record the same seismic trace at the source position. The acquisition pattern is expanded using the principle of reciprocity as depicted in Figure .

 

ideal-acq
Figure 1 A fixed split-spread acquisition that is ideal for wavefront synthesis imaging.

 

real-acq
Figure 2 A one-side spread acquisition, which is the typical acquisition pattern in a marine survey.

 

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Figure 3 The expanded acquisition pattern from the one-side spread using reciprocity. There are missing regions in the near and far offset.

However, Figure  shows that we still have some missing regions in near and far offset after the acquisition expansion that uses the principle of reciprocity. The missing part of the far-offset limits the angle coverage of the reflectivity information. Since the maximum incidence angle of the source wave field generally decreases as the depth of reflector increases (Figure ), we can avoid the effects of cable length by synthesizing different incidence angle ranges at every depth level.

The effects of the missing part of the near offset are not as easily avoided as those of the missing part of the far offset, because the effect of missing near-offset traces differ in terms of missing reflection angle ranges from depth to depth and are located around the normal incidence angle to reflectors (Figure ). In order to obtain a good image in every constant angle illumination, we need to interpolate the missing near offset traces.

 

ang-cov
Figure 4 Ray paths are depicted for a CMP gather. The effects of missing near-offset traces appear in different places according to the dips of reflectors.